City Research Online

Abstract

For quantum computers to become useful tools to physicists, engineers and computational scientists, quantum algorithms for solving nonlinear differential equations need to be developed. Despite recent advances, the quest for a solver that can integrate nonlinear dynamical systems with a quantum advantage, while being realizable on available (or near-term) quantum hardware, is an open challenge. In this paper, we transform a nonlinear dynamical system into a linear system, which we integrate with quantum algorithms. Key to the method is the Fokker–Planck equation, which is a non-normal partial differential equation. Three approaches are proposed: (i) forward-Euler stepping by block encoding; (ii) Schrödingerization and (iii) forward-Euler stepping by a linear combination of unitaries. We emulate the integration of prototypical nonlinear systems with the proposed quantum solvers and compare the output with the benchmark solutions of classical integrators. We find that classical and quantum outputs are in good agreement. This paper opens opportunities for solving nonlinear differential equations with quantum algorithms.

Publication Type:	Article
Additional Information:	© 2025 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Publisher Keywords:	quantum algorithms, quantum computing, nonlinear systems, applied quantum computation
Subjects:	Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments:	School of Science & Technology School of Science & Technology > Department of Engineering
SWORD Depositor:	Symplectic Administrator

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